Here \(\nu _1, \nu _2, \rho _1, \rho _2\)  are the kinematic viscosities and the densities of the two fluids in the bulk, respectively. Also, \(f_{i, eq}^k(\mathbf {x},t)\)  is the \(i^{th}\)  equilibrium discrete probability distribution function for the \(k^{th}\)  component, formally derived as a low Mach, second-order expansion of a Maxwellian probability equilibrium distribution [1]}
\( f_{i, eq}^k(\mathbf {u},\mathbf {x},t)=w_i \rho _k \Bigg (1+3 \mathbf {c}_i \cdot \mathbf {u} + 9 \frac{(\mathbf {c}_i \cdot \mathbf {u})^2}{2} -3 \frac{\mathbf {u} \cdot \mathbf {u}}{2} \Bigg ),\) 
